11.2 Limits

Limits For functions of a single variable, when we let x approach a, there are only two possible directions of approach, from the left or from the right. If the limits from these two directions are different, then the function itself has no limit as x approaches a.

as long as (x, y) remains in the domain of f For functions of two variables the situation is more complicated because (x, y) can approach (a, b)… from an infinite number of directions, in any manner whatsoever, as long as (x, y) remains in the domain of f

Does not exist Thus, if we can find two different paths of approach along which the function has different limits, then f(x, y) has no limit as (x, y) approaches (a, b).

Introduction We want to compare the behaviors of as both x and y approach 0. The next two slides show tables of values of f(x, y) and g(x, y) for points (x, y) near the origin:

show that does not exist: First we approach (0, 0) along the x-axis: f(x, 0) = x2/x2 = 1 for all x ≠ 0, so f(x, y) approaches 1 as (x, y) approaches (0, 0) along the x-axis. Next we approach (0, 0) along the y-axis: f(0, y) = –y2/y2 = –1 for all y ≠ 0, so f(x, y) approaches –1 as (x, y) approaches (0, 0) along the y-axis. Since f has two different limits along two different lines, the given limit does not exist.

If f(x, y) = xy/(x2 + y2), does exist? Solution Again we consider various paths: f(x, 0) = 0/x2 = 0 for all x ≠ 0, so f(x, y) approaches 0 as (x, y) approaches (0, 0) along the x-axis. f(0, y) = 0/y2 = 0 for all y ≠ 0, so f(x, y) approaches 0 as (x, y) approaches (0, 0) along the y-axis as well.

Although we have obtained identical limits along the axes, that does not show that the given limit is 0. Let’s now approach (0, 0) along another line, say y = x: For all x ≠ 0, -Therefore f(x, y) approaches ½ as (x, y) approaches (0, 0) along y = x. -Since we have obtained different limits along different paths, the given limit does not exist, as the next slide illustrates.